To clarify, by random games you mean just any games we can think of? Or do you mean actually 'random games' like roguelikes?
If the latter (or even if not I suppose) there's one that infuriates me, but I can never really deny my attraction for. Two huge names in indie gaming created quite a phenomenal test of torture. It is programmed and designed by Terry Cavanagh, (VVVVVV, Don't Look Back) with art by Annabelle Kennedy (Cave Story Anthem, myriad indie art to die for). Also some ridiculously incredible music by Dubmood (never heard of 'em, sorry). The resulting amazingiousness was titled SELF DESTRUCT!
If you'd like to know why, feel free to attempt a place on the top 50 online highscore table, of which the creator Terry tops out at number 45! Minimum entry is well over 2,000,000 points! GOOD LUCK!!
Also, didn't know if you meant browser games too, it's a download but hey. It's great.